Necessary Conditions for Optimality in Problems with Nonstandard Cost Functionals

Part of the Systems Control: Foundations & Applications book series (SCFA)


Usual formulation of optimal control problems involves the minimization of a cost functional which is of the form of a definite integral. In this chapter we develop necessary conditions for an optimal control in the case of problems in which the cost functional is either a quotient or a product of definite integrals. We call such functionals nonstandard, and these naturally arise in Chapters 2,4, and 5 during the computation of the performance of a suboptimal H∞ controller. In Chapters 2,4, and 5 a criterion for the evaluation of the cost functional will be presented in the specialized case of linear systems and quadratic integrands. Preliminary results for problems having a fixed final time and free terminal state are in [1]. Related results can also be found in [2,3]. In this monograph, we consider only fixed final time problems. Problems in which the final time is free are treated in [4]. In Section 5, we discuss the relation of our results to those in [2,3].


Optimal Control Problem Final Time Tangent Direction Dual Cone Finite Horizon 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Subrahmanyam, M. B. and Eyman, E. D., “Optimization with non-standard cost functional,” Proc. 13th Annual Allerton Conference, University of Illinois, Urbana, IL, 1975, pp. 168–173.Google Scholar
  2. [2]
    Miele, A., “The extremization of products of powers of functional and its application to aerodynamics,” Astronautica Acta, 12, No. 1, 1967, pp. 47–51.Google Scholar
  3. [3]
    Miele, A., “On the minimization of the product of the powers of several integrals,” Journal of Optimization Theory and Applications, 1, No. 2, 1967, pp. 70 - 82.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    Subrahmanyam, M. B., “Necessary conditions for minimum in problems with nonstandard cost functionals,” Journal of Mathematical Analysis and Applications, 60, No. 3, 1977, pp. 601–616.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    Dubovitskii, A. YA. and Milyutin, A. A., “Extremum problems in the presence of restrictions” [English translation], U.S.S.R. Computational Mathematics and Mathematical Physics, 5, No. 3, 1965, pp. 1–80.zbMATHCrossRefGoogle Scholar
  6. [6]
    Girsanov, I. V., Lectures on Mathematical Theory of Extremum Problems, Lecture Notes in Economics and Mathematical Systems, No. 67, Springer-Verlag, Berlin, 1972.Google Scholar
  7. [7]
    Köthe, G., Topological Vector Spaces, Vol. I, Springer-Verlag, Berlin, 1969.zbMATHGoogle Scholar

Copyright information

© Birkhäuser Boston 1995

Authors and Affiliations

  1. 1.Flight Dynamics and Control Branch Air Vehicle & Crew Systems Technoly Dept.Naval Warfare Center Aircraft DivisionWarminsterUSA

Personalised recommendations